The affine yangian \(\mathcal {Y}\)  has a representation on the plane partitions. A plane partition \(\pi \)  is a 2D Young diagram in the first quadrant of plane \(xOy\)  filled with non-negative integers that form nonincreasing rows and columns [1]}, [2]}. The number in the position \((i,j)\)  is denoted by \(\pi _{i,j}\)
\(\left( \begin{array}{ccc}\pi _{1,1} & \pi _{1,2} &\cdots \\\pi _{2,1} & \pi _{2,2} &\cdots \\\cdots & \cdots &\cdots \\\end{array} \right).\) 
